Finite Group Databasebeta

← #27272655

Subgroups — #27272655

Order 13568.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283066C211yesmaximal, characteristic
2#20274782C211yesmaximal, characteristic
3#20274476C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1271C211yesmaximal, characteristic, complemented
8C2×#13133216C411yescharacteristic
9C2×#13133195—21no—
10C2×C42×D53C2211yescharacteristic
11C2×#13133129—21no—
12C2×#13133129—21no—
13C2×#13133127—21no—
14#16810527—21no—
15#16810521C2211yescharacteristic
16#16810518—21no—
17#16810492C411yescharacteristic
18#16810482—21no—
19#16810480C411yescharacteristic
20#16810480C411yescharacteristic
21#16810378C2211yescharacteristic
22#16810378C2211yescharacteristic
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106—21no—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×#88322—21no—
32C22×#88322—21no—
33#13133216—42no—
34#13133195—42no—
35#13133195—42no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C42×D53—42no—
48#13133129—24no—
49#13133129—42no—
50#13133127—42no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54C22×C8×C53—21no—
55C22×C8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C42×C53C2311yescharacteristic
60#95×C53—21no—
61C23×D53C22⋊C411yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C2×C4×D53—22no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×Dic106—42no—
85C2×Dic106—42no—
86C2×Dic106—42no—
87C2×C8×C53—22no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C42×C53—41no—
99#11719921—42no—
100#11719921—42no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53—41no—
104#11719919—42no—
105#11719919—42no—
106#11719919—41no—
107#11719919—41no—
108#11719919—42no—
109C4×Dic53—22no—
110C4×Dic53—22no—
111C4×Dic53—41no—
112C2×#88322—22no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C23×C53C2×C22⋊C411yescharacteristic
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53#10112yes—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53#13111yescharacteristic
139C2×C4×C53#13111yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53#13711yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53#9912yes—
158C2×Dic53#10412yes—
159C4×D53—42no—
160C4×D53—42no—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—42no—
165Dic106—84no—
166C8×C53—42no—
167C8×C53—81no—
168#88322—42no—
169#88322—81no—
170#7705—531nomaximal, Sylow
171C22×C53#40811yescharacteristic
172C22×C53C2×#9911yescharacteristic
173C22×C53C2×#10111yescharacteristic
174C22×C53#41811yescharacteristic
175C22×C53#41911yescharacteristic
176C22×C53C2×#10411yescharacteristic
177C22×C53#42111yescharacteristic
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—81no—
192C4×C53—81no—
193Dic53—22no—
194Dic53—22no—
195Dic53—41no—
196Dic53—81no—
197Dic53—81no—
198#2857—531no—
199#1817—531no—
200#1749—531no—
201#1271—531no—
202#1271—531no—
203#1271—531no—
204#1271D5311yescharacteristic, complemented
205C2×C53#147211yescharacteristic
206C2×C53#148511yescharacteristic
207C2×C53#149611yescharacteristic
208C2×C53#150711yescharacteristic
209C2×C53#151111yescharacteristic
210C2×C53#153111yescharacteristic
211C2×C53#155611yescharacteristic
212D53—42no—
213C2×C4⋊Q8—531no—
214C2×C22⋊Q8—1061no—
215C22×C42—531no—
216C2×C22⋊C8—1061no—
217C2×C22⋊C8—1061no—
218C2×#97—1061no—
219#391—1061no—
220#389—531no—
221#389—1061no—
222#389C2×D5311yescharacteristic
223#384—531no—
224#380—531no—
225#380—531no—
226#380—1061no—
227#336—531no—
228#336—531no—
229#336—1061no—
230#336C2×D5311yescharacteristic
231#336C2×D5311yescharacteristic
232C53#770511yescharacteristic, complemented, Sylow
233C22×Q8—1061no—
234C23×C4—531no—
235C23×C4—531no—
236C23×C4—531no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—1061no—
240C22×C8—1061no—
241C4⋊Q8—2122no—
242C22⋊Q8—2122no—
243C22⋊Q8—2122no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4C4×D5311yescharacteristic
252C2×C4⋊C4C4×D5311yescharacteristic
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C42—531no—
256C2×C42—531no—
257C2×C42—531no—
258C2×C42—2122no—
259C2×C42C22×D5311yescharacteristic
260C22⋊C8—1064no—
261C22⋊C8—2122no—
262#97—2122no—
263#95—21no—
264#95—1061no—
265#95—1061no—
266#95—1061no—
267C24—531no—
268C2×Q8—2122no—
269C2×Q8—2122no—
270C2×Q8—2122no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—1062no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4C2×C4×D5311yescharacteristic
291C22×C4D4×D5311yescharacteristic
292C22×C4D4×D5311yescharacteristic
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—1062no—
300C2×C8—2121no—
301C2×C8—2121no—
302C2×C8—2121no—
303C2×C8—2121no—
304C2×C8—2121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—2121no—
313C4⋊C4—2122no—
314C4⋊C4—2122no—
315C4⋊C4—2122no—
316C4⋊C4—2121no—
317C4⋊C4—2121no—
318C22⋊C4—2122no—
319C22⋊C4—2122no—
320C42—41no—
321C42—1062no—
322C42—1062no—
323C42—2121no—
324C23—532no—
325C23—1061no—
326C23—1061no—
327C23—1061no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23C22⋊C4×D5311yescharacteristic
332Q8—4244no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4#1313322311yescharacteristic
367C2×C4#1313322511yescharacteristic
368C2×C4#1313324111yescharacteristic
369C2×C4#1313324411yescharacteristic
370C2×C4#1313324611yescharacteristic
371C2×C4#1313324911yescharacteristic
372C8—42no—
373C8—81no—
374C8—4241no—
375C8—2122no—
376C22—1062no—
377C22—1062no—
378C22—1062no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22#1681058811yescharacteristic
384C22#99×D5311yescharacteristic
385C22#101×D5311yescharacteristic
386C22#1681062611yescharacteristic
387C22#1681063011yescharacteristic
388C22#104×D5311yescharacteristic
389C22#1681063811yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—1062no—
399C4—1062no—
400C4—4241no—
401C4—4241no—
402C4—2121no—
403C2—2122no—
404C2#2027307711yescharacteristic
405C2#2027313411yescharacteristic
406C2#2027318211yescharacteristic
407C2#2027322911yescharacteristic
408C2#2027324611yescharacteristic
409C2#2027332811yescharacteristic
410C2#2027343011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7705
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #20272539C2semidirect
5. #20272538C2semidirect
6. #20272537C2semidirect
7. C53×#1271C2semidirect
204. #1271D53semidirect
232. C53#7705semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202830662
#202747822
#202744762
#202725392
#202725382
#202725372
C53×#12712
#770553